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Display information for equation id:math.1665.20 on revision:1665

* Page found: Drehimpuls- Eigenzustände (eq math.1665.20)

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'''\begin{align}

& {{{\hat{L}}}_{+}}{{{\hat{L}}}_{-}}=\left( {{{\hat{L}}}_{1}}+i{{{\hat{L}}}_{2}} \right)\left( {{{\hat{L}}}_{1}}-i{{{\hat{L}}}_{2}} \right)={{{\hat{L}}}_{1}}^{2}+{{{\hat{L}}}_{2}}^{2}-i\left[ {{{\hat{L}}}_{1}},{{{\hat{L}}}_{2}} \right]={{{\hat{L}}}^{2}}-{{{\hat{L}}}_{3}}^{2}+\hbar {{{\hat{L}}}_{3}} \\

& {{{\hat{L}}}_{-}}{{{\hat{L}}}_{+}}={{{\hat{L}}}_{1}}^{2}+{{{\hat{L}}}_{2}}^{2}+i\left[ {{{\hat{L}}}_{1}},{{{\hat{L}}}_{2}} \right]={{{\hat{L}}}^{2}}-{{{\hat{L}}}_{3}}^{2}-\hbar {{{\hat{L}}}_{3}} \\

& \to \left[ {{{\hat{L}}}_{+}},{{{\hat{L}}}_{-}} \right]=2\hbar {{{\hat{L}}}_{3}} \\

& \left[ {{{\hat{L}}}^{2}},{{{\hat{L}}}_{+}} \right]=0 \\

& \left[ {{{\hat{L}}}^{2}},{{{\hat{L}}}_{-}} \right]=0 \\

\end{align}

TeX (checked): 

'''{\begin{aligned}&{{\hat {L}}_{+}}{{\hat {L}}_{-}}=\left({{\hat {L}}_{1}}+i{{\hat {L}}_{2}}\right)\left({{\hat {L}}_{1}}-i{{\hat {L}}_{2}}\right)={{\hat {L}}_{1}}^{2}+{{\hat {L}}_{2}}^{2}-i\left[{{\hat {L}}_{1}},{{\hat {L}}_{2}}\right]={{\hat {L}}^{2}}-{{\hat {L}}_{3}}^{2}+\hbar {{\hat {L}}_{3}}\\&{{\hat {L}}_{-}}{{\hat {L}}_{+}}={{\hat {L}}_{1}}^{2}+{{\hat {L}}_{2}}^{2}+i\left[{{\hat {L}}_{1}},{{\hat {L}}_{2}}\right]={{\hat {L}}^{2}}-{{\hat {L}}_{3}}^{2}-\hbar {{\hat {L}}_{3}}\\&\to \left[{{\hat {L}}_{+}},{{\hat {L}}_{-}}\right]=2\hbar {{\hat {L}}_{3}}\\&\left[{{\hat {L}}^{2}},{{\hat {L}}_{+}}\right]=0\\&\left[{{\hat {L}}^{2}},{{\hat {L}}_{-}}\right]=0\\\end{aligned}}

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L̂+L̂−=(L̂1+iL̂2)(L̂1−iL̂2)=L̂12+L̂22−i[L̂1,L̂2]=L̂2−L̂32+ℏL̂3L̂−L̂+=L̂12+L̂22+i[L̂1,L̂2]=L̂2−L̂32−ℏL̂3→[L̂+,L̂−]=2ℏL̂3[L̂2,L̂+]=0[L̂2,L̂−]=0‴
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data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">−</mo><mi>i</mi><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><msup><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><msup><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">−</mo><mi>i</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo>,</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><msup><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">−</mo><msup><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><mi alternate="1">ℏ</mi><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mo stretchy="false" lspace="0" rspace="0">−</mo></mrow></msub><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mo stretchy="false" lspace="0" rspace="0">+</mo></mrow></msub><mo stretchy="false">=</mo><msup><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><msup><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><mi>i</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo>,</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><msup><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">−</mo><msup><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">−</mo><mi alternate="1">ℏ</mi><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false" accent="false">→</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mo stretchy="false" lspace="0" rspace="0">+</mo></mrow></msub><mo>,</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mo stretchy="false" lspace="0" rspace="0">−</mo></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><mn>2</mn><mi alternate="1">ℏ</mi><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msup><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>,</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mo stretchy="false" lspace="0" rspace="0">+</mo></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msup><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>,</mo><msub><mover><mi>L</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mo stretchy="false" lspace="0" rspace="0">−</mo></mrow></msub><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo stretchy="false">=</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable><mo>&#x2034;</mo></msup></mstyle></mrow></math>

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Identifiers

  • L̂+
  • L̂−
  • L̂1
  • i
  • L̂2
  • L̂1
  • i
  • L̂2
  • L̂1
  • L̂2
  • i
  • L̂1
  • L̂2
  • L̂
  • L̂3
  • L̂3
  • L̂−
  • L̂+
  • L̂1
  • L̂2
  • i
  • L̂1
  • L̂2
  • L̂
  • L̂3
  • L̂3
  • L̂+
  • L̂−
  • L̂3
  • L̂
  • L̂+
  • L̂
  • L̂−

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